You can find the answer to this question in the Epilogue. There, on page 411 in the paperback edition of the book, Diamond says that the Fertile Crescent lost its head start because it was "an ecologically fragile environment."
The Fertile Crescent was a great place for agriculture to get started. However, it was not very well-suited to agriculture on a large scale. As the civilizations in the area got bigger, they put more and more stress on their environments. Today, Diamond says, it is "absurd" to think of Southwest Asia as a "Fertile Crescent." The land has been overused and can no longer sustain enough agriculture to feed a large population. Because the Fertile Crescent was unlucky in its geography, it was not able to sustain large populations and it lost its initial head start.